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How is Crul wrong in anything other than the terminology? You sign a document with your private key - generating basically a hash of the document entangled with your key information. Anyone holding the public key can then verify that hash with the public key - that the document contents are intact and unchanged (from the hash), and generated by the person holding the private key (entangled key information)
Thanks for mediating!
What I'm getting from this dicussion is that, when signing, the operations are not encryption and decryption, but ... hashing and hash-veryfing?
To help you with the terminology, the names for the two operations are "signing" and "verifying". That's it.
What can you do with...
"Signing" is not at all the same as "encrypting" with the keys swapped. It is a separate specific sequence of mathematical operations you perform to combine two numbers (the private key and the message) to produce a third - the signature. Signing is not called "hashing". A hash may be involved as part of the signature process, but it is not strictly necessary. It makes the "message" number smaller, but the algorithm can sign the full message without hashing it first, will just require computation for longer time. "Hash-verifying" isn't a thing in this context, you made that name up, just use "verify".
@dohpaz42 is mad because you messed up your terminology originally, and thought you were trying to say that you "encrypt" a message with the private key, which is totally backwards and wrong. He didn't know that in your mind you thought you were talking about "signing" the message. Because honestly no one could have known that.